Methods for improved heading estimation

ABSTRACT

Methods for calibrating a body-worn magnetic sensor by spinning the magnetic sensor 360 degrees to capture magnetic data; if the spin failed to produce a circle contained in an x-y plane fit a sphere to the captured data; determining offsets based on the center of the sphere; and removing the offsets that are in the z-direction. Computing a magnetic heading reliability of a magnetic sensor by determining an orientation of the sensor at one location; transforming the orientation between two reference frames; measuring a first vector associated with the magnetic field of Earth at the location; processing the first vector to generate a virtual vector when a second location is detected; measuring a second vector associated with the magnetic field of Earth at the second location; and calculating the magnetic heading reliability at the second location based on a comparison of the virtual vector and the second vector.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a divisional of U.S. patent application Ser. No.13/915,998, filed Jun. 12, 2013, which claims benefit under 35 U.S.C. §119(e) of Provisional U.S. Patent Application No. 61/783,908, filed Mar.14, 2013, and of Provisional U.S. Patent Application No. 61/658,883filed Jun. 12, 2012, the contents of which are incorporated herein byreference in their entireties.

GOVERNMENT RIGHTS

This invention was made with government support under W91CRB-09-C-0089,awarded by the US Army, and W31P4Q-12-C-0043, awarded by the DefenseAdvanced Research Projects Agency. The Government has certain rights inthe invention.

TECHNICAL FIELD

The disclosure relates to systems and methods for locating, trackingand/or monitoring the status of personnel and/or assets, both indoorsand outdoors, and more particularly in GPS denied environments.

SUMMARY

Methods are disclosed for calibrating in real-time a magnetic sensorthat include at least the steps of spinning the magnetic sensor 360degrees to capture magnetic data along the 360 degrees spun path;determining that the spinning failed to produce a circle that iscontained in an x-y plane; fitting a sphere to the captured magneticdata; determining offsets based on the center of the sphere; andremoving the offsets that are in the z-direction. Systems are disclosedfor computing an indicator of a magnetic heading reliability of amagnetic sensor, including at least a processor; a memorycommunicatively coupled to the processor, and instructions causing thecomputing system to at least determine an orientation of the magneticsensor at a first location; generate a filter that transforms theorientation between two reference frames; measure a first vectorassociated with the magnetic field of Earth at the first location; whena second location of the magnetic sensor is detected, process the firstvector through the filter to generate a virtual vector; measure a secondvector associated with the magnetic field of Earth at the secondlocation; and calculate the indicator of the magnetic headingreliability at the second location based on a comparison of the virtualvector and the second vector.

BRIEF DESCRIPTION OF THE FIGURES OF THE DRAWING

FIG. 1 is an illustration of an embodiment of location data pointsgenerated during a 360 degree spin of a magnetic sensor producing adeformed circle;

FIG. 2 is an illustration of an embodiment of uncalibrated andcalibrated location data points from the sensor fit to a sphere;

FIG. 3A illustrates a magnetic field description of a geographiclocation;

FIG. 3B is an illustration of three path locations near the geographiclocation of FIG. 3A;

FIG. 4 is an illustration of an embodiment of a plot of magnetic fieldmagnitude for a first location among the three path locations of FIG.3B;

FIG. 5 is an illustration of an embodiment of a plot of magnetic fieldinclination for the first location of FIG. 4;

FIG. 6 is an illustration of an embodiment of a plot of the computerheading output for the first location of FIG. 4;

FIG. 7 is an illustration of an embodiment of a plot of magnetic fieldmagnitude for a second location among the three path locations of FIG.3B;

FIG. 8 is an illustration of an embodiment of a plot of magnetic fieldinclination for the second location of FIG. 7;

FIG. 9 is an illustration of an embodiment of a plot of the computerheading output for the second location of FIG. 7;

FIG. 10 is an illustration of an embodiment of a plot of magnetic fieldmagnitude for a third location among the three path locations of FIG.3B;

FIG. 11 is an illustration of an embodiment of a plot of magnetic fieldinclination for the third location of FIG. 10;

FIG. 12 is an illustration of an embodiment of a plot of the computerheading output for the third location of FIG. 10;

FIG. 13 is an illustration of an embodiment of reference frames for asensor worn by a user;

FIG. 14 is an illustration of an embodiment of a plot of virtualmagnetic sensor data for the third location of FIG. 3B;

FIG. 15 is an illustration of an embodiment of a plot of the actualmagnetic sensor data, for a calibrated body frame of reference, for thethird location of FIG. 14;

FIG. 16 is an illustration of an embodiment of a plot of virtualmagnetic sensor data for the first location of FIG. 3B;

FIG. 17 is an illustration of an embodiment of a plot of the actualmagnetic sensor data, for a calibrated body frame of reference, for thefirst location of FIG. 16;

FIG. 18 is an illustration of an embodiment of a plot of the fusionangle for each of the locations of FIG. 3B;

FIG. 19 is an illustration of an embodiment of a plot of a true pathversus an inertial path estimate with uncompensated bias;

FIG. 20 is an illustration of an embodiment of a plot of a high driftpath versus an inertial path estimate with uncompensated bias;

FIG. 21 is an illustration of an embodiment of a plot comparing a gyroheading, a compass heading and compass reliability for an example path;

FIG. 22 is an illustration of an embodiment of a plot comparing a gyrominus compass heading and transformed data for the example path of FIG.21;

FIG. 23 is an illustration of an embodiment of a plot comparing a firstpath and a drift corrected version of the first path where entropydecreases;

FIG. 24 is an illustration of an embodiment of a plot comparing a gyrominus compass heading and a drift estimate for the path of FIG. 23;

FIG. 25 is an illustration of an embodiment of a plot comparing a secondpath and a drift corrected version of the second path where entropyincreases;

FIG. 26 is an illustration of an embodiment of a plot comparing a gyrominus compass heading and a drift estimate for the path of FIG. 25;

FIG. 27 is an illustration of an embodiment of a plot comparing a thirdpath and a drift corrected version of the third path where entropyincreases;

FIG. 28 is an illustration of an embodiment of a plot comparing a gyrominus compass heading and a drift estimate for the path of FIG. 27;

FIG. 29 is an illustration of an embodiment of a plot comparing a fourthpath and a drift corrected version of the fourth path where entropyincreases;

FIG. 30 is an illustration of an embodiment of a plot comparing a gyrominus compass heading and a drift estimate for the path of FIG. 29;

FIG. 31 is an illustration of an embodiment of spatial gyro and compassheading data weighted overalaid on an original patent and a rotatedinertial path;

FIG. 32 is an illustration of an embodiment of rotation and scale limitsdue to constraints; and

FIG. 33 is an illustration of an embodiment of a computer system.

DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS

To increase the achievable accuracy of systems that provide locationsindoors or in other GPS denied environments, the computation of anaccurate heading is a critical element. An estimate of globallyreferenced heading is necessary to initialize an inertial system'sstarting heading and for occasional heading drift correction. Magneticmeasurements are extremely valuable in navigation systems for providinga global referenced heading that is not subject to the accumulation oferror. However, several factors may cause the heading from a magneticcompass to become unreliable. These include sensor calibration errors,local sources of interference, and interference caused by thesurrounding environment.

Embodiments disclosed herein are directed to techniques for the run timecalibration of magnetic sensors, such as body-worn sensors, that enablelocal sources of interference to be calibrated out after mounting.Embodiments disclosed herein are also directed to techniques forcomputing an indicator of magnetic heading reliability, techniques ofusing the compass heading data to remove gyro heading drift, techniquesfor global rotation estimation using spatial magnetometer filtering, andtechniques for limiting the solution space of constraint based onheading optimization.

Method for Body-Worn Magnetic Sensor Calibration

Techniques have been developed for performing calibration of magneticsensors, such as the magnetic sensors in body worn devices or devicesattached to some form of mobile object. There are typically two sourcesof calibration error in magnetometers: local disturbances from sourcesof flux rotating along with the sensor, called hard-iron errors, andenvironmental distortions to the earth's flux in the earth's coordinateframe, called soft-iron errors. Hard-iron errors are constant andmanifest themselves in an offset of the magnetometer's output, and cantherefore be compensated for with a one-time calibration. The normalprocedure for calibration in 3-axes is to collect a series of readingswhile the magnetometer is rotated about all three axes. A sphere is fitto this data, and the center of this sphere determines the offsets.

For a mobile tracking system based on a mounted tracking unit, localdisturbances are caused by the device's electronics and can also becaused by other items on the same platform. For example, if the deviceis mounted on a person, what the person is wearing or carrying with themcould affect the sensor calibration. Because of this, it is mosteffective to calibrate the device once mounted, but at that point it maybe difficult to rotate the device around all axes.

For example, in a personnel tracking system, the compass is used todetermine a heading in the x-y earth frame so calibration in this planeis the most important. In order to calibrate the magnetometer inreal-time while it is attached to the body, a person can spin 360degrees, to produce a circle of location data points, which are mostlyin the x-y earth frame. It is difficult for a person to remain perfectlyin the level frame during such a spin, so a circle fit of the locationdata points in the x-v plane will often fail because the z-component isnot constant, which produces deformed circles, such as illustrated inFIG. 1. In accordance with an embodiment herein, instead of attemptingto fit the location data points to a circle, the data points are fit toa sphere, which produces good results in the x and y directions, butill-conditioned results in the z direction. In such a case, the z-offsetmay be ignored while the x and y offsets (relative to the center of thesphere) are applied to fit the location data points toward the surfaceof the sphere, thereby calibrating the sensor. An example ofuncalibrated and calibrated location data points in accordance with theabove are illustrated in FIG. 2, where the uncalibrated result is shownabove the calibrated result.

Method for Computing an Indicator of Magnetic Heading Confidence

In buildings, magnetic compass data is often affected by large magneticdisturbances due to structural and electromagnetic sources of magneticfield. However, there are also areas where the heading data is good. Itis important to have an indication of when the compass heading isreliable so that it can be used for occasional reference when navigatingin buildings and other GPS denied environments.

In order to provide a confidence level for magnetic field vectormeasurements, it must first be determined what a valid measurementshould look like. There are several features of the Earth's magneticfield that are helpful for distinguishing that magnetic field from othersources. These include field magnitude, field stability, and magneticinclination or “dip” angle. The magnetic inclination is the anglebetween the Earth's, magnetic field vector and the horizon (level plane)at a specific location. While each of these values slowly change overtime, they may be considered constant for short (i.e., months)durations. The magnitude of the field measured on the Earth's surfacevaries according to location, but is often nominally considered to beabout 0.5 gauss (50 micro Tesla). Declination for a given locationdescribes the angle between magnetic North and true North. Because thedeclination value cannot be measured unless true North is known, it willnot be very useful for determining field accuracy.

An experiment was performed in an attempt to find indicators that willhelp determine the validity of each heading measurement taken by asensor. The premise of the experiment was that if a level of confidencewith each heading measurement could be reliably reported, then it mightbe possible to more accurately make heading corrections. For theexperiment, it was assumed that the sensor calibration was accurate andthat compensation had been made for all local sources of error.

The experiment assumed that the location of the sensor was fixed, inthis case in Greenbelt, Md., and that the magnetic sensor's calibrationhad been normalized such that the Earth's magnetic field would report avalue of ˜1.0 units. FIG. 3A of the Drawing provides a magnetic fielddescription for the Greenbelt, Md. location, based on data from theNational Oceanic and Atmospheric Administration Geophysical Data Centerat the time of the experiment. As noted, Greenbelt, Md. is located at amagnitude of about 52.2 uT and an inclination of about 66.5 degrees.Declination at Greenbelt, Md. is about 10.8 degrees W.

The magnitude was calculated directly from the magnetic field sample.The inclination was determined by comparing the magnetic field vector tothe vector as determined by the gyros and accelerometers sensors of anInertial Navigation Unit (INU) worn by a subject. The field stabilitywas calculated as a function of field strength and orientation comparedto changing user location. Lastly, a comparison between magnetic fieldorientation and device orientation was used for validating the results.

Because field stability and sensor agreement metrics are only valid whenthe device bearing the sensor is moving, they will not contribute to theheading confidence result when the user is stopped.

Three test locations were chosen so that each may be used to demonstratea different level of compass angle accuracy. As illustrated in FIG. 3B,while raw data was collected outdoors for the experiment, no publicdisclosure or use in public of any embodiment disclosed herein resultedfrom such outside activities. The first location, labeled “1) MTC1 path”in FIG. 3B, was along a line inside of an office building, on the 8^(th)floor, where there were many source of severe interferences. The secondlocation, labeled “2) East Corner Path” in FIG. 3B, was along a linenear the east corner of a parking lot in front of the office building.The third location, labeled “3) South Corner Path” was along a line nearthe sour corner of the same parking lot.

At each location, the subject wearing the sensor walked 20-35 m along astraight path with a known heading of 340.8 (330+10.8), then stopped,turned around 180 degrees, and walked back to the start location with aknown heading of 160.8 (150+10.8). Calibration was applied to raw datalogs and data was transformed into navigation frame coordinates, whichcompensates for tilt. Field intensity, compass heading, and magneticinclination were calculated for each test as follows:

$\begin{matrix}{{{M}} = \sqrt{m_{x}^{2} + m_{y}^{2} + m_{z}^{2}}} & {{Equation}\mspace{14mu} 1} \\{{Heading} = {{{\tan^{- 1}\left( \frac{- m_{y}}{m_{x}} \right)} \star \left( \frac{180}{\pi} \right)} - 90}} & {{Equation}\mspace{14mu} 2} \\{{Inclination} = {\sin^{- 1}\left( \frac{m_{z}}{{M}} \right)}} & {{Equation}\mspace{14mu} 3}\end{matrix}$

The results for the three test locations are summarized in FIGS. 4-12,where FIGS. 4, 7 and 10 correspond to the first location, FIGS. 5, 8 and11 correspond to the second location, and FIGS. 6, 9 and 12 correspondto the third location. Each plots illustrated in these figures are formagnetic field magnitude and field inclination along with the computedheading for each given location. The undistorted earth magnetic fieldmagnitude has been normalized to be 1, as shown in the FIGS. 4, 7 and10. The undistorted earth magnetic field inclination of 66.5 degrees isindicated in FIGS. 5, 8 and 11. The true headings of 340.8 degrees, then160.8 are indicated in the FIGS. 6, 9 and 12. An examination of the dataillustrated in these plots reveals that the tests with the least errorin field intensity (magnitude), FIGS. 5 and 6, and field inclination,FIGS. 8 and 9, generally produced a more accurate heading output, FIGS.11 and 12.

Field Based Reliability Measure

The techniques for determining static reliability reflect the generalfinding that the compass is most accurate when magnitude and inclinationmeasurements match their expected values. If the magnitude orinclination is within some percent of its expected value, then it isconsidered reliable. Outside of that range, the reliability is decreasedas a function of the error. Listing 1 below gives an example of a sourcecode implementation that assumes that the magnetic field magnitude hasbeen normalized to 1.

Listing 1 float getStaticReliability(float3x1 mag) { //CalculateMagnitude and Inclination. float magnitude =sqrt(float3x1_dot(mag,mag)); float inclination =asin(abs(mag.z)/magnitude) * 180/M_PI; //Calculated difference betweenmeasured and expected values. float mError = abs(magnitude−1.0); floatiError = abs(inclination−LOCAL_MAG_INCLINATION); //Normalize error tovalue between 0 and 1. //Assume data is unreliable outside of 0.5magnitude and 25 inclination errors. if(mError>0.5) mError = 1; elsemError = mError/0.5; if(iError>25) iError = 1; else iError = iError/25;//Note: if both measurments are as expect 1 is returned, indicatinghighest confidence. return (1−mError)*(1−iError); }Virtual Orientation Sensor Based Reliability Measure

In addition to the sensors, the Inertial Navigation Unit also containsan orientation filter algorithm that determines the orientation of thedevice based on input from the gyroscope and accelerometer sensors. Theoutput of this filter is a quaternion that represents the transformationthat is required to map vectors from one reference frame to another.With respect to a human subject, the “body frame's” coordinate systemhas its +x-axis pointing outward from the right side of the user's bodyand its +y-axis pointing outward from the user's stomach. The sensor'sraw output is in body frame coordinates. As illustrated in FIG. 13, the“earth frame's” coordinates system has its +y-axis pointing horizontallytoward the direction the device was facing when it was first turned on.Its +z-axis is pointing upwards directly opposite of gravity. Lastly,the “navigation frame's”+y-axis points the direction the user is facingbut remains parallel to the horizon and its +z-axis points upwards,directly opposite of gravity. Each reference frame uses a right handedcoordinate system.

In order to determine how the magnetic field vector is supposed toideally move if its output were to match that of the orientation filter,an embodiment may utilize a virtual magnetic sensor output, such asillustrated in FIG. 14, where the x-axis is identified as 1402, they-axis is identified as 1404, and the z-axis is identified as 1406. Todo this, a vector was first defined that matched that of the magneticfield at the first sample of a test path. This vector was thentransformed by the orientation filter output quaternion at eachsubsequent sample and the resulting vector was recorded. Assuming theorientation filter's output has zero error, this vector should match theoutput of an ideal magnetic sensor in a magnetic field with zerointerference. As illustrated in FIG. 15, for the calibrated body frame,the resulting vector is plotted along with the data measured by theactual magnetic sensor for the third location of the experiment notedabove, which was the path with the least error. In FIG. 15, the x-axisis identified as 1502, the y-axis as 1504 and the z-axis as 1506. Acomparison of these results show that 1052 is very close to 1402, 1504is very close to 1404 and 1506 is very close to 1406, which means thatthe virtual sensor is incredibly accurate at predicting the magneticsensor output, which in turn indicates that the magnetic data from thesensor is likely valid. Note that the “virtual sensor” has no input fromthe magnetic field sensor except for its very first point and themagnetic sensor data shown has no input from the gyro or accelerometersensors. In comparison, FIG. 16 is a plot of the virtual sensor for thefirst location of the experiment noted above, which was the path withthe greatest error, where the x-axis is 1602, the y-axis is 1604 and thez-axis is 1606, while FIG. 17 is a plot of the calibrated body frame forthe same path, where 1702 is the x-axis, 1704 is the y-axis and 1706 isthe z-axis. The noted mismatch between these two plots indicates thatthe magnetic data should not be trusted as valid.

For the conducted tests described above, the angular heading results forthe virtual sensor were very accurate and equal to or more accurate thanthe magnetic sensor. This was expected over the short term because whencalculating heading, this technique is equivalent to that of integratingthe rate gyro with its start value set to the correct heading. However,this cannot be relied upon in general because there are many situationswhere the first sample from the magnetic sensor will not be free ofinterference. Instead, it is necessary to slowly work to get an estimateof the correct magnetic heading over time.

As such, a feedback technique was developed that allows the “virtual”magnetic vector to move slowly over time toward the measured magneticvector whenever the wearer is moving. If the wearer stops moving, thedifference between the virtual magnetic vector and measured magneticvector will be fixed, but not necessarily correct, so it is importantnot to allow this to move towards zero. The difference between themeasured and virtual magnetic field vectors at each sample is calculatedand serves as an indicator of how well the magnetic field vector istracking the orientation of the device.

The heading calculated from the “virtual” sensor vector is mostlycontrolled by the inertial orientation filter output, but is slowly“pulled” toward the measured magnetic field vector whenever the user ismoving. This heading is referred to as the “fusion” angle because it isa measure of the angle between the virtual sensor vector and themeasured results. FIG. 18 illustrates the fusion angle for each of thethree test data sets, which test 1 is identified as 1802, test 2 isidentified as 1804 and test 3 is identified as 1806.

From FIG. 18, it can be seen that the fusion angle is really an errormeasurement indicator of magnetic field reliability (low error, e→highreliability, r). Any function that maps this error information onto aweighting space is valuable for reducing the effect of poor heading datawhile improving the effect of good heading data. An example reliabilityfunction is defined as follows:

$r\left\{ \begin{matrix}{100 - e^{2}} & {e < 10} \\0 & {e \geq 10}\end{matrix} \right.$Method for Drift Estimation and Removal

A gyro has small amounts of bias that cannot be corrected bycalibration. This turn-on bias is randomly produced at initialization.There is also a lower level of in-run drift that could be caused bytemperature variation, but for purposes of the present disclosure, biasis assumed to be fairly constant over the short term. This bias ispresented in the computed path as drift. For example, in FIG. 19 theeffect of uncompensated z-axis gyro bias on a path estimate isillustrated, where the true path is 1902 and the uncompensated rawcomputed path is 1904. For purposes of the present disclosure, thisdrift is referred to herein as inertial drift.

Despite the inertial drift that is apparent over a longer period oftime, over a very short period of time, the gyro is accurate. In thisregard, “short” is relative to the quality of the gyroscope, wherein lowcommercial grade gyros drift rather quickly relative to military,navigation and tactical grade gyros. For example, while the gyro headingmay drift over a long straight path, a gyro is typically a goodindicator for turning, since turns occurs over a relatively short time.A magnetic sensor, on the other hand, provides angle with absolutereference in presence of a “clean” Earth field, but if the compasscalibration is not good, a turn may result in the compass heading andgyro heading changing by different amounts; the larger the change inheading, the larger the difference will be. With these issues in mind,compass/gyro fusion algorithms can be developed to use the bestproperties of each sensor to eliminate drift and allow the system tomaintain good absolute heading accuracy.

When the magnetic field data is undisturbed and the sensors are properlycalibrated, small changes in inertial heading should match the changesin magnetic field heading. Despite environmental disturbances inmagnetic field measurements, for persons or assets moving around in theenvironment (even in magnetically poor environments, such as inbuildings), the relationship between the compass readings and gyro rateestimates can usually be discovered. Eliminating the gyro bias enableslonger duration GPS denied tracking.

As disclosed herein, a technique has been developed for drift estimationand removal that accounts for issues that may be encountered whencompass calibration may be poor. An example illustrated in FIG. 20demonstrates the output of various processing steps based on a path withsignificant drift to exaggerate the issues. FIG. 20 shows path data 2002with compensation for drift and path data 2004 without driftcompensation, where the drift is approximately −0.37 degrees/second.FIG. 21 shows the sensor (gyro and compass) heading angle andreliability data for the example path illustrated in FIG. 20 over a 30minute period. The gyro heading is identified as a series of smallsquare markings on the graph generally enclosed within dashed lines2102. The compass reliability is also identified as a series of smallsquare markings (largely grouped between 0 and 100) on the graph, whichare generally enclosed within the dashed-dotted lines 2104. The compassheading is identified as the other small square markings 2106 on thegraph.

In this example, the focus was on removing drift in the z-axis gyrousing navigation frame magnetic heading, but the technique applies todrift removal in any axis. The technique has the following steps:

Step 1: Based on the “compass reliability” indication described above,filter out compass data with low reliability.

Step 2: Transform the gyro data and compass data into the navigationframe and compute compass heading minus the gyro heading, Which isillustrated as 2202 in FIG. 22. The slope of this line, gyro minuscompass (Gyro-Compass) 2202, can be attributed to drift.

Step 3: Compute and remove the average drift. To do this, best fit linesmay be computed using y=mx+b, were m is the slope and b is the interceptmod 360. This may be accomplished by a variety of techniques. In theillustrated implementation, the Hough transform is used to find thelines. Using the average slope, and intercept across all lines, thedrift is subtracted off to leave the transformed data 2204, as shownnear the bottom of the graph in FIG. 22. The transformed data 2204should be approximately linear. In the case where the compasscalibration is poor, the transformed data would have lines with similarslope but with varying offset (not an issue in this example).

Step 4: To account for improper compass calibration, the compass datamay be sorted into bins based on compass angle. For each bin, a weightedbest fit line for compass minus gyro 2202 is computed. The weight in thecomputation is based on the compass reliability, such as compassreliability 2104 from FIG. 21. The slope for each of these linesrepresents an adjustment to the computed drift from the previous step.The drift may vary slightly from bin to bin. The median slope, m₂,represents the drift adjustment and the median average deviationrepresents the accuracy of the computation. The final driftestimate=m₁+m₂.

The standard deviation of the transformed data, σ_(m), provides ameasure of potential drift variation.

In performing the steps set forth above for removing drift in any axis,several implementation considerations may need to be considered. First,when the posture of the person or asset changes, the z-axis gyro in thenavigation frame uses a different physical gyro. This different gyro hasa different drift rate. The technique can be restarted when significantorientation changes are detected to separate drift calculationsdepending on posture.

Second, sudden errors in gyro angle (for example, caused by gyro ratesaturation) will cause an offset in the gyro-compass plot at the time ofthe saturation. Thus, the line will no longer be continuous. Thesesaturations of the gyro can be detected and the algorithm restarted toremedy this.

Third, drift computation is most accurate when the device is very still.This is because when the device is still, drift can be computedindependent of any variations in the compass (such as poor calibration)and independent of variations in the magnetic field. A method has beenimplemented to detect a strict “no motion” condition by requiring thatthe accelerometer and gyroscope variations remain below a certainthreshold. Then, if no motion is maintained over a certain period oftime the drift can be precisely computed by computing the bias termssuch that the gyro computed headings remain constant over the sameperiod. The resulting heading drift can be computed from the slope ofthe best fit line of computed heading over the period, m_(s) which willbe approximately 0. The variation of the data from 0, σ_(s), can be usedfor predicting possible heading errors.

Fourth, when the user is wearing the device, periods of walking may beinterspersed with periods of relative stillness. As pointed out above,because this computation is independent of the compass, during theseperiods of stillness, accurate gyro drift can be computed even in poormagnetic field conditions if these periods of stillness last beyond athreshold. Drift computed during periods of stillness should be moreaccurate than drift computed during motion, so this drift computationcould be used to set/reset drift or this term could be very heavilyweighted in a weighted average drift computation relative to driftcomputed as a function of the gyro-compass heading during motion. Forexample, the final drift might be m₁+m₂+Wm where m₁+m₂ are driftscomputed during motion, m_(s) is computed while the user is still, andthe weight W>>1.

Lastly, extreme magnetic variations can cause the system to calculateincorrect drift rates. Techniques for filtering magnetic data asdiscussed in the section above entitled “Method for Computing anIndicator of Magnetic Heading Confidence” may be used to minimize theeffect of magnetic disturbances. If incorrect estimates are computed,they are generally unstable and can be detected by comparing driftcalculations that have been computed in succession; calculate drift,apply drift, recalculate drift, obtain a new drift, and so on. If thenew drift is not a small number, then the original drift estimate waspoor. If after several drift estimates, the drift estimation does notapproach zero, then the calculations are too scattered to be reliableand should not be used. The above described drift compensation techniquehas been able to correct drift of up to 0.75 degrees per second. Ifthere are poor magnetic conditions, it may not be able to estimatedrift, but as time progresses the estimation capability may improve, aslong as there are some areas with a good or fairly good magnetic field.

Drift Engine Integrity Check via Pseudo Entropy

In an embodiment, a pseudo entropy computation is utilized to provide aconfidence in the drift number described in the above drift estimationor any other drift estimation technique. In accordance with thisembodiment, when drift is compensated, this generally increases “order”in the path (for example, segments overlap) and thereby reduces entropy.

It is often clear from visual inspection of a path whether the driftcorrections “look good” even without having access to the ground truth.Pseudo entropy provides a means for quantifying good drift correction,versus bad drift correction, with respect to the original path withouthaving access to ground truth. The pseudo entropy is defined by:E=(w1*Path Volume+w2*Path Occupancy)/(Path Length)+w3*(Line Factor)where w1, w2, and w3 are weighting constants which are respectively 10,20, and 1/20, in this embodiment.

The first term rewards paths that overlap. The Path Volume is the convexhull of the set of path points. The Path Occupancy is computed by takinga fixed volume around each path point (for example 1 m³ in thisembodiment) and then computing the union of the volumes for the path.The Path Length is the length of the path.

The second term rewards paths that have straight segments. The LineFactor can be implemented in a variety of ways. For example, straightlines could be rewarded by performing a linear regression on the set ofpath points within some time before and after the current path point (30seconds in this embodiment) and then summing the residual errors overthis set of points from the line. This can be done for select pathpoints and the number errors summed. In this embodiment, the computationmay be done at each path point. Another embodiment may reward straightlines grouped into fewer heading directions. This embodiment may beimplemented using the Hough transform (a standard transform in imageprocessing for finding dominant lines in images). The number of dominantlines found with different angles is used as the line factor.

An example of the pseudo entropy solution is illustrated in FIG. 23,where path 2302 represents an original path with an entropy of 7.7897.The lower entropy solution of path 2304, having an entropy of 7.3882,has fewer dominant headings. FIG. 24 further illustrate the same pathexample, showing the gyro minus compass difference as 2402 and the gyroerror estimate 2404 with entropy reduces as expected. A second exampleis illustrated in FIGS. 25 and 26. In FIG. 25, an original sample pathis shown as 2502 without drift correction and an entropy of 9.3507, andthe same path 2504 with drift correction and an entropy lowered to6.7861, In FIG. 26, the compass difference 2602 and the drift estimatefor the same path are shown with entropy reduced as expected. The firsttwo examples illustrate a fairly significant amount of path overlap aswell as strait path segments, which are rewarded by the pseudo entropyengine's algorithm.

Two different examples are illustrated in FIGS. 27-30, in which thedrift engine fails to improve the drift. As show in FIG. 27, theoriginal path 2702 is shown on the left and the drift corrected path2704 is on the right. Entropy for path 2702 was 5.0985, but increased to6.6354. As can be expected, the pseudo entropy check based the compassdifference 2802 and drift estimate for the same path, shown in FIG. 28,reflects the same, and as a result the drift computations may berejected. Likewise, in FIG. 29, the original path 2902 is shown on theleft and the drift corrected path 2904 is shown on the right, withentropy increases from 3.0706 to 4.3850. The entropy check illustratedin FIG. 30 shows that the applied drift decreases tracking performancewhile the entropy value was increased. Consequently, drift correctionwas again rejected.

Method for Global Rotation Estimation using Spatial MagnetometerFiltering

Because magnetic sensor data can be poor in certain locations, a spatialmagnetic filter has also been developed to provide an additional measureof protection from having a location with poor magnetic datadisproportionally affect the computed heading. For example, a path mayhave segments or portions in areas with some magnetic field distortion,but other portions or segments in areas with an undistorted field. Thisis a common scenario, for example, going to the mall or the office—inthis scenario people walk a short distance from their cars to thebuilding, where there are areas of significant distortion, Even if thepoor sensor data has reduced weight because of the computed magneticfield confidence, if each measurement is allowed to contribute otherwiseequally to the final heading, then the weight of the contribution fromthat a location becomes a function of the time the user spends at thatlocation. This could be problematic.

To better handle this issue, a spatial filter has been developed thatdistributes the weight of the magnetic field contribution equally over aspatial grid. FIG. 31 shows an example of spatially distributed compassand gyro heading data overlaid on the inertial path. The left side ofFIG. 31 shows the raw gyro and compass heading vector data, which isweighted by the computed magnetic heading confidence described above.Notice that the confidence is low for much of the data inside thebuilding and even near the building. The right side of FIG. 31 shows thepath with the heading computed using the spatial filter described below.In this side, the final compass-gyro heading vector is shown withoutweighting. For portions of the path with good heading information, thisvector should have 0 heading (indicated by pointing to the right).

An example implementation of this technique is as follows:

Step 1: Each grid location may be assigned only one heading. In thisimplementation a 0.5 meter grid is used. The weighted average of themagnetic heading minus the gyro heading is computed for all pointswithin the 0.5 meter square, G_(xy), is assigned to the grid location.The weight is reflective of the magnetic reliability as computed above.Note that with drift removed and with a well calibrated magnetometer,the magnetic headings minus the gyro heading should be near zero.

$\left( {c - g} \right)_{x,y} = {\sum\limits_{{k❘x},{y \in G_{x,y}}}{{r(k)}^{\star}\left( {{c(k)} - {g(k)}} \right)}}$note that the weighted vectorr(k)*(c(k)−g(k))is a vector of length between 0 and 1.

Step 2: The average of spatially distributed data is computed and a gyroinitial rotation is found that minimizes this value. The rotation thatminimizes the average of spatially distributed magnetic heading minusthe gyro heading represents the final best guess for global rotation ofthe inertial path.

$\min\limits_{g{(0)}}{\sum\limits_{x,y}\left( {c - g} \right)_{x,y}}$

Step 3: An error measure on the global heading estimate is provided bythe weighted standard deviation of the spatially distributed headings,σ_(R). This can be used to limit the amount the solvers can rotate thepath from this initial estimate.

$\sigma_{R} = \sqrt{- {\ln\left( {\sum\limits_{x,y}{\left( {{c(k)} - {g(k)}} \right)_{x,y}/\left( {\sum\limits_{x,y}{r\text{/}{\sum\limits_{x,y}1}}} \right)}} \right)}}$Method for Limiting the Solution Space of Constraint Based Optimization

Constraints on the navigation solution are obtained from GPS, ranging toanchors, user corrections and check-ins, etc. The navigation system usesconstraint based optimization to compute a globally optimal solution forlocation initialization, rotation, path scale, and drift by tweakingthese four parameters. Initialization of the drift and rotation arecomputed for example as described above. Limits are placed on the amountthat parameters are allowed to change.

There are several ways to limit the rotation of the path. As describedabove, the weighted standard deviation of compass values may becalculated. The rotation is limited by ±nσ_(R), where n=3 is used togive a bound which contains the heading about 99% of the time assumingapproximately Gaussian distribution. This is more important when themagnetic field is clean and the compass computed heading is good. Inthis case, it enforces tight bounds on possible rotations. The headingwill likely not have Gaussian distribution in the case where there issignificant magnetic field distortion; however, in that case, σ_(R) willbe large and so the bound will not impose much of a restriction asdesired.

Additionally, as illustrated in FIG. 32, which shows two equivalenttechniques for finding the angle bounds, if an inertial path travels adistance D in a straight line with endpoints bounded by constraints withradius R₁ and R₂, then rotation is bounded by analyzing the worst caseangles of lines that are tangent to both constraint circles, as follows:2 arcsin((R ₁ +R ₂)/D)

As for scale, if an inertial path travels a distance D in a straightline with endpoints bounded by constraints with radii R₁ and R₂, thenglobal scale bounds are determined by computing:(D+R ₁ +R ₂)/(Length of Inertial Segment)and(D−R ₁ −R ₂)/(Length of Inertial Segment).

Drift is constrained to be within the bounds of the devicespecification. It can also be constrained to lie within +nσ_(m), where nis an integer. n=3 has been used, requiring the drift to lie withinthree times the standard deviation of the computed drift, as notedabove.

For constraints that are not met, computation of a local modification ofthe path to satisfy these remaining constraints is completed using aphysics based particle solver.

Embodiments described herein include a method for calibrating inreal-time a body-worn magnetic sensor, the method comprising capturingmagnetic data from the magnetic sensor when the sensor moves along a 360degree spun path, if the captured magnetic data fails to produce acircle contained in an x-v plane, fitting a sphere to the capturedmagnetic data, determining x-axis, y-axis and z-axis offsets for thecaptured magnetic data fit to the sphere based on a center of thesphere, removing the z-axis offsets, and using the x-axis offsets andy-axis offsets to calibrate the magnetic sensor.

Embodiments described herein include a computing system for computing anindicator of a magnetic heading reliability of a magnetic sensor, thecomputing system comprising a processor, a memory communicativelycoupled to the processor, the memory bearing instructions that, whenexecuted on the processor, cause the computing system to at least,determine an orientation of the magnetic sensor at a first location,generate a filter that transforms the orientation between two referenceframes, measure a first vector associated with the magnetic field ofEarth at the first location, when a second location of the magneticsensor is detected, process the first vector through the filter togenerate a virtual vector, measure a second vector associated with themagnetic field of Earth at the second location, and calculate theindicator of the magnetic heading reliability at the second locationbased on a comparison of the virtual vector and the second vector.

The embodiment described herein, wherein the computing system comprisesthe magnetic sensor.

The embodiment described herein, wherein the first vector associatedwith the magnetic field comprises magnitude, stability, and inclinationvalues of the magnetic field of Earth at the first location.

Embodiment described herein include a computer readable storage mediumcomprising instructions that, when executed on a computing systemconfigured to calculate a gyro heading drift associated with a magneticsensor and a gyro sensor, cause the computing system to at leasttransform gyro data and compass data into a navigation frame, compute acompass heading and a gyro heading in the navigation frame, determine afirst drift based on a best fit line of the compass heading and the gyroheading in the navigation frame, sort the compass data into bins basedon compass angle, determine a second drift based on an average slope ofbest fit lines of the gyro data subtracted from the compass data in eachbin, and calculate the gyro heading drift based on the first drift andthe second drift.

The embodiment described herein, wherein the computing system comprisesthe magnetic sensor and the gyro sensor.

The embodiment described herein, wherein the best fit lines aregenerated based on a Hough transform.

The embodiment described herein, Wherein a slope of a best fit line isattributed to drift.

The embodiment described herein, wherein, for each bin, the best fitline is weighted based on a compass reliability.

The embodiment described herein, wherein the instruction further causethe computer system to at least compute a confidence in the gyro headingdrift based on entropy of a path taken by the magnetic sensor and thegyro sensor, maintain the gyro heading drift if entropy decreases, andreject the gyro heading drift if entropy increases.

The embodiment described herein, wherein the confidence is based onoverlap in the path and straight segments in the path.

Embodiment described herein include a global rotation estimationobtained by the steps of determining a path of a magnetic sensor andgyro sensor in motion in an environment based on magnetic data receivedfrom the magnetic sensor and gyro data received from the gyro sensor,breaking the path into a plurality of grids, for each grid, computing aheading based on magnetic headings and gyro headings at locations withinthe path in each grid, determining an average heading across theplurality of grids, and setting the global rotation estimation as avalue that minimizes the average heading.

The embodiment described herein, wherein the heading of a grid of theplurality of grids is a weighted average of the magnetic headings andgyro headings at the locations within the path in the grid.

The embodiment described herein, wherein the steps further compriseapplying an error measure to the average heading such that globalrotation estimation accounts for the error.

The embodiment described herein, wherein the error comprises a standarddeviation of the headings of the plurality of grids.

The techniques described above can be implemented on a computing deviceassociated with a user (e.g., gyroscope and accelerometer sensorsimplemented on a device worn by the user), a plurality of computingdevices associated with a plurality of users, a server in communicationwith the computing device(s) (e.g., a server configured to calibrate thegyroscope and accelerometer sensors of the device worn by the user), ora plurality of servers in communication with the computing device(s).Additionally, the techniques may be distributed between the computingdevice(s) and the server(s). For example, the computing device maycollect and transmit raw data to the server that, in turn, process theraw data to improve heading estimation. The following figure describes acomputing system that includes hardware modules, software module, and acombination thereof and that can be implemented as the computing deviceand/or as the server.

In a basic configuration, as illustrated in FIG. 33, the computingsystem 3300 may include at least a processor 3302, a system memory 3304,a storage device 3306, input/output peripherals 3308, communicationperipherals 3310, and an interface bus 3312. The interface bus 3312 isconfigured to communicate, transmit, and transfer data, controls, andcommands between the various components of the computer system, such asan electronic device. The system memory 3304 and the storage device 3306comprise computer readable storage media, such as RAM, ROM, EEPROM,hard-drives, CD-ROMs, optical storage devices, magnetic storage devices,flash memory, and other tangible storage media. Any of such computerreadable storage medium can be configured to store instructions orprogram codes embodying aspects of the disclosure. Additionally, thesystem memory 3304 comprises an operation system and applications. Theprocessor 3302 is configured to execute the stored instructions and cancomprise, for example, a logical processing unit, a microprocessor, adigital signal processor, and the like.

The system memory 3304 and the storage device 3306 may also comprisecomputer readable signal media. A computer readable signal medium mayinclude a propagated data signal with computer readable program codeembodied therein. Such a propagated signal may take any of variety offorms including, but not limited to, electro-magnetic, optical, or anycombination thereof. A computer readable signal medium may be anycomputer readable medium that is not a computer readable storage mediumand that can communicate, propagate, or transport a program for use inconnection with the computing system 3300.

Further, the input and output peripherals 3308 include user interfacessuch as a keyboard, screen, microphone, speaker, other input/outputdevices, and computing components such as digital-to-analog andanalog-to-digital converters, graphical processing units, serial ports,parallel ports, and universal serial bus. The input/output peripherals3308 may be connected to the processor through any of the ports coupledto the interface bus.

The user interfaces can be configured to allow a user of the computingsystem 3300 to interact with the computing system 3300. For example, thecomputing system 3300 may include instructions that, when executed,cause the computing system 3300 to generate a user interface that theuser can use to provide input to the computing system 3300 and toreceive an output from the computing system.

This user interface may be in the form of a graphical user interfacethat is rendered at the screen and that is coupled with audiotransmitted on the speaker and microphone and input received at thekeyboard. In an embodiment, the user interface can be locally generatedat the computing system 3300. In another embodiment, the user interfacemay be hosted on a remote computing system (not shown but accessiblethrough communication peripherals 3310, and rendered at the computingsystem 3300. For example, the server may generate the user interface andmay transmit information related thereto to the computing device 3300that, in turn, renders the user interface to the user. The computingdevice 3300 may, for example, execute a browser or an application thatexposes an application program interface (API) at the server to accessthe user interface hosted on the server.

Finally, the communication peripherals 3310 of the computing system 3300are configured to facilitate communication between the computing system3300 and other computing systems (e.g., between the computing device andthe server) over a communications network. The communication peripherals3312 include, for example, a network interface controller, modem,various modulators/demodulators and encoders/decoders, wireless andwired interface cards, antenna, and the like.

The communication network includes a network of any type that issuitable for providing communications between the computing device andthe server and may comprise a combination of discrete networks which mayuse different technologies. For example, the communications networkincludes a cellular network, a WiFi/broadband network, a local areanetwork (LAN), a wide area network (WAN), a telephony network, afiber-optic network, or combinations thereof. In an example embodiment,the communication network includes the Internet and any networks adaptedto communicate with the Internet. The communications network may be alsoconfigured as a means for transmitting data between the computing deviceand the server.

The techniques described above may be embodied in, and fully orpartially automated by, code modules executed by one or more computersor computer processors. The code modules may be stored on any type ofnon-transitory computer-readable medium or computer storage device, suchas hard drives, solid state memory, optical disc, and/or the like. Theprocesses and algorithms may be implemented partially or wholly inapplication-specific circuitry. The results of the disclosed processesand process steps may be stored, persistently or otherwise, in any typeof non-transitory computer storage such as, e.g., volatile ornon-volatile storage.

The various features and processes described above may be usedindependently of one another, or may be combined in various ways. Allpossible combinations and sub-combinations are intended to fall withinthe scope of this disclosure. In addition, certain method or processblocks may be omitted in some implementations. The methods and processesdescribed herein are also not limited to any particular sequence, andthe blocks or states relating thereto can be performed in othersequences that are appropriate. For example, described blocks or statesmay be performed in an order other than that specifically disclosed, ormultiple blocks or states may be combined in a single block or state.The example blocks or states may be performed in serial, in parallel, orin some other manner. Blocks or states may be added to or removed fromthe disclosed example embodiments. The example systems and componentsdescribed herein may be configured differently than described. Forexample, elements may be added to, removed from, or rearranged comparedto the disclosed example embodiments.

Conditional language used herein, such as, among others, “can,” “could,”“might,” “may,” “e.g.,” and the like, unless specifically statedotherwise, or otherwise understood within the context as used, isgenerally intended to convey that certain embodiments include, whileother embodiments do not include, certain features, elements, and/orsteps. Thus, such conditional language is not generally intended toimply that features, elements and/or steps are in any way required forone or more embodiments or that one or more embodiments necessarilyinclude logic for deciding, with or without author input or prompting,whether these features, elements and/or steps are included or are to beperformed in any particular embodiment. The terms “comprising,”“including,” “having,” and the like are synonymous and are usedinclusively, in an open-ended fashion, and do not exclude additionalelements, features, acts, operations, and so forth. Also, the term “or”is used in its inclusive sense (and not in its exclusive sense) so thatwhen used, for example, to connect a list of elements, the term “or”means one, some, or all of the elements in the list.

While certain example embodiments have been described, these embodimentshave been presented by way of example only, and are not intended tolimit the scope of the inventions disclosed herein. Thus, nothing in theforegoing description is intended to imply that any particular feature,characteristic, step, module, or block is necessary or indispensable.Indeed, the novel methods and systems described herein may be embodiedin a variety of other forms; furthermore, various omissions,substitutions and changes in the form of the methods and systemsdescribed herein may be made without departing from the spirit of theinventions disclosed herein. The accompanying claims and theirequivalents are intended to cover such forms or modifications as wouldfall within the scope and spirit of certain of the inventions disclosedherein.

What is claimed:
 1. A method for obtaining a global rotation estimation,comprising: determining a path of a magnetic sensor and a gyro sensor ina device in motion in an environment based on magnetic data receivedfrom the magnetic sensor and gyro data received from the gyro sensor,the path comprising a plurality of path points; segmenting the path suchthat path points are separated into a plurality of spatial grids;determining magnetic headings from the magnetic data and gyro headingsfrom the gyro data for the path points in each spatial grid among theplurality of spatial grids; for each spatial grid, computing a gridvalue based on a function of the magnetic headings and the gyro headingsat path points within each spatial grid; determining an average of thegrid values across the plurality of spatial grids; and setting theglobal rotation estimation as a rotation value that minimizes theaverage of the grid values.
 2. The method of claim 1, wherein there aretwo or more path points in a spatial grid, and wherein the grid value isa weighted average of the magnetic headings and the gyro headings at thetwo or more path points.
 3. The method of claim 1, further comprisingapplying an error measure to the average of the grid values such thatthe global rotation estimation accounts for the error measure.
 4. Themethod of claim 3, wherein the error measure comprises a standarddeviation of the grid values of the plurality of spatial grids.
 5. Acomputer-implemented method for spatially weighting a path constraint,comprising: receiving sensor data from a tracked computing device, thesensor data including magnetic data from a magnetic sensor and gyro datafrom a gyro sensor; determining a magnetic heading based on the magneticdata and a gyro heading based on the gyro data; determining a path ofthe tracked computing device, the path comprising a plurality of pathpoints based on the magnetic heading and the gyro heading; segmentingthe path such that path points are separated into a plurality of spatialgrids; for each spatial grid among the plurality of spatial grids,computing a grid value based on a function of the magnetic headings andthe gyro headings at path points within each spatial grid; determiningan average grid value across the plurality of spatial grids; and settingthe path constraint as a constraint value that minimizes the averagegrid value.
 6. The method of claim 5, wherein the path constraint is arotation, a path scale, or a drift.
 7. The method of claim 5, whereinthe grid value is a weighted average of a difference between themagnetic heading and the gyro heading at the path points within eachspatial grid.
 8. The method of claim 5, further comprising applying anerror measure to the average grid value to decrease error in the pathconstraint.
 9. The method of claim 8, wherein the error measurecomprises a standard deviation of the grid values of the plurality ofspatial grids.
 10. A computer readable storage medium comprisinginstructions that, when executed on a computing system of a trackeddevice having a magnetic sensor for a compass and a gyro sensor, causethe computing system to: receive sensor data comprising magnetic datafrom the magnetic sensor and gyro data from the gyro sensor; determine amagnetic heading based on the compass data and a gyro heading based onthe gyro data; determine a path of the tracked computing device, thepath comprising a plurality of path points based on the magnetic headingand the gyro heading; segment the path such that path points areseparated into a plurality of spatial grids; for each spatial grid amongthe plurality of spatial grids, compute a spatial grid heading based ona function of the magnetic headings and the gyro headings at path pointswithin each spatial grid; determine an average grid value across theplurality of spatial grids; and set a path constraint as a constraintvalue that minimizes the average grid value.
 11. The computer readablestorage medium of claim 10, wherein the path constraint is a rotation, apath scale, or a drift.
 12. The computer readable storage medium ofclaim 10, wherein the grid value is a weighted average of a differencebetween the magnetic heading and the gyro heading at the path pointswithin each spatial grid.
 13. The computer readable storage medium ofclaim 10, wherein the instructions, when executed on the processor,further cause the computer system to at least apply an error measure tothe average grid value to decrease error in the path constraint.
 14. Thecomputer readable storage medium of claim 13, wherein the error measurecomprises a standard deviation of the grid values of the plurality ofspatial grids.
 15. A computing system comprising: a processor; a memorycommunicatively coupled to the processor, the memory bearinginstructions that, when executed on the processor, cause the computingsystem to: receive sensor data from a tracked computing device, thesensor data including magnetic data from a magnetic sensor and gyro datafrom a gyro sensor; determine a magnetic heading based on the magneticdata and a gyro heading based on the gyro data; determine a path of thetracked computing device, the path comprising a plurality of path pointsbased on the magnetic heading and the gyro heading; segment the pathsuch that path points are separated into a plurality of spatial grids;for each spatial grid among the plurality of spatial grids, compute agrid value based on a function of the magnetic headings and the gyroheadings at path points within each spatial grid; determine an averagegrid value across the plurality of spatial grids; and set a pathconstraint as a constraint value that minimizes the average grid value.16. The system of claim 15, wherein the grid value is a weighted averageof a difference between the magnetic heading and gyro heading at pathpoints within the spatial grid.
 17. The system of claim 15, wherein theinstructions, when executed on the processor, further cause thecomputing system to at least apply an error measure to the average gridvalue to decrease error in the path constraint.
 18. The system of claim17, wherein the error measure comprises a standard deviation of the gridvalues of the plurality of spatial grids.